To convert the polar equation r = 3 cos(8θ) into a Cartesian equation, we can use the relationships between polar and Cartesian coordinates:
- x = r cos(θ)
- y = r sin(θ)
- r = √(x² + y²)
- cos(θ) = x/r
- sin(θ) = y/r
Starting with the polar equation:
r = 3 cos(8θ)
Multiplying both sides by r gives:
r^2 = 3r cos(8θ)
Now, substituting the identities:
r^2 = x² + y² and r cos(θ) = x
We know that:
cos(8θ) = cos^8(θ) + 8cos^7(θ)sin(θ) + 28cos^6(θ)sin^2(θ) + 56cos^5(θ)sin^3(θ) + 70cos^4(θ)sin^4(θ) + 56cos^3(θ)sin^5(θ) + 28cos^2(θ)sin^6(θ) + 8cos(θ)sin^7(θ) + sin^8(θ)
However, for simplification, we will find cos(8θ) directly in terms of x and y. Using the identity r^2 = x^2 + y^2, we can write:
3cos(8θ) = 3(x/r)^8 + other terms
But a more straightforward approach is noting that cos(8θ) represents an 8-fold symmetry. Hence:
The Cartesian transformation gives
x^2 + y^2 = 3x
Ultimately, we arrive at:
Cartesian Equation: x^2 + y^2 = 3x
This equation describes a circle with a center at (3/2, 0) and a radius of √(9/4) = 3/2 centered horizontally with the symmetry corresponding to cos(8θ).
Thus, the curve described by the polar equation r = 3 cos(8θ) is an 8-petaled rose curve.